Question

(a) If your utility is represented by u(x; y) = min(x+2y; 2x+y);what
do your indi¤erence curves look like?
(b) Given your answer in (a), obtain the MRS (marginal rates of sub-
stitution).
(c) Suppose the prices of x and y are px = $3 and px = $1 and you
have 100 dollars. What would you choose?
(d) If px decreases to $1; what would you choose?
(e) Use the Slutsky decomposition to decompose the total price e¤ect
into the substitution e¤ect and income e¤ect when px decreases
from $3 to $1:

Answer 1

a) The utility function U= min(X+2y, y+2x) is L shaped curve. Min utility function depict perfect complements with a kink where both goods are consumed in fixed proportions.

b) MRS for a L shaped utility function is infinte at the vertical part and zero at the horizontal part as explained in the diagram below. The MRS is not defined at the kink

c) Budget constraint: Px X+Py.Y =M

For given question 3X +Y = 100

Optimal bundle is X+2Y = Y+2X

Solving we get, X=Y

Substituting in budget constrain we get, 3X+X= 100

4X= 100, X=100/4 = 25

Therefore, consumer chooses X=Y=25, since goods are perfect complements, both commodities are consumed in equal proportion to get maximum utility.

d) New budget constrain when Px =$1, X+Y = 100

Since, X=Y, X+X=100, 2X=100

Or X =100/2 = 50

The new bundle is X=Y = 50, the consumer consumes higher amount of both goods as price oc one good decreases because they are perfect complements.